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Original Citation:
Big Bang 6Li nucleosynthesis studied deep underground (LUNA collaboration)
Published version:
DOI:10.1016/j.astropartphys.2017.01.007
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Contents
lists
available
at
ScienceDirect
Astroparticle
Physics
journal
homepage:
www.elsevier.com/locate/astropartphys
Big
Bang
6
Li
nucleosynthesis
studied
deep
underground
(LUNA
collaboration)
D.
Trezzi
a
,
M.
Anders
b
,
c
,
1
,
M.
Aliotta
d
,
A.
Bellini
e
,
D.
Bemmerer
b
,
A.
Boeltzig
f
,
g
,
C.
Broggini
h
,
C.G.
Bruno
d
,
A.
Caciolli
h
,
i
,
F.
Cavanna
e
,
P.
Corvisiero
e
,
H.
Costantini
e
,
2
,
T.
Davinson
d
,
R.
Depalo
h
,
i
,
Z.
Elekes
b
,
M.
Erhard
h
,
F.
Ferraro
e
,
A.
Formicola
f
,
Zs.
Fülop
j
,
G.
Gervino
k
,
A.
Guglielmetti
a
,
C.
Gustavino
l
,
∗
,
Gy.
Gyürky
j
,
M.
Junker
f
,
A.
Lemut
e
,
3
,
M.
Marta
b
,
4
,
C.
Mazzocchi
a
,
5
,
R.
Menegazzo
h
,
V.
Mossa
m
,
F.
Pantaleo
m
,
P.
Prati
e
,
C.
Rossi
Alvarez
h
,
D.A.
Scott
d
,
E.
Somorjai
j
,
O.
Straniero
n
,
o
,
T.
Szücs
j
,
M.
Takacs
b
a Università degli Studi di Milano and INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy b Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany c Technische Universität Dresden, Mommsenstrasse 9, 01069 Dresden, Germanyd SUPA, School of Physics and Astronomy, University of Edinburgh, EH9 3JZ Edinburgh, United Kingdom e Università degli Studi di Genova and INFN, Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy f Laboratori Nazionali del Gran Sasso (LNGS), Via Acitelli 22, 67010 Assergi (AQ), Italy
g Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, 67100 L’Aquila (AQ), Italy h INFN, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy
i Dipartimento di Fisica e Astronomia, Università degli studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy j Institute for Nuclear Research (MTA ATOMKI), PO Box 51, HU-4001 Debrecen, Hungary
k Università degli Studi di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy l INFN, Sezione di Roma Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy
m Dipartimento Interateneo, Università degli Studi di Bari and INFN, Sezione di Bari, Via E. Orabona 4, 70125 Bari, Italy n Osservatorio Astronomico di Collurania, Via M. Maggini, 64100 Teramo, Italy
o INFN, Sezione di Napoli, Via Cintia, 80126 Napoli, Italy
a
r
t
i
c
l
e
i
n
f
o
Article history:
Received 20 November 2015 Revised 2 December 2016 Accepted 22 January 2017 Available online 23 January 2017
Keywords:
Nuclear Astrophysics Big Bang Nucleosynthesis 6Li abundance Underground
a
b
s
t
r
a
c
t
Thecorrectpredictionoftheabundancesofthelightnuclides producedduringtheepochofBigBang Nucleosynthesis (BBN) is one ofthe main topics ofmodern cosmology. For manyof the nuclear re-actions thatarerelevant forthisepoch, directexperimental crosssection dataareavailable, ushering the so-called“ageof precision”.The present work addresses an exception tothis currentstatus: the 2 H(
α
,γ
)6 Lireactionthatcontrols6 LiproductionintheBigBang.Recentcontroversialobservationsof6 Li inmetal-poorstarshaveheightenedtheinterestinunderstandingprimordial6 Liproduction.Ifconfirmed, theseobservationswouldleadtoasecondcosmologicallithiumproblem,inadditiontothewell-known 7 Liproblem. Inthepresentwork,thedirectexperimentalcrosssection dataon2 H(α
,γ
)6 LiintheBBN energyrangearereported.ThemeasurementhasbeenperformeddeepundergroundattheLUNA (Lab-oratoryforUndergroundNuclearAstrophysics)400kVacceleratorintheLaboratoriNazionalidelGran Sasso,Italy.ThecrosssectionhasbeendirectlymeasuredattheenergiesofinterestforBigBang Nucle-osynthesisforthefirsttime,atEcm =80,93,120,and133keV.Basedonthenew data,the 2 H(α
,γ
)6 Li thermonuclearreactionratehasbeenderived.Ourrateisevenlowerthanpreviouslyreported,thus in-creasingthediscrepancybetweenpredictedBigBang6 Liabundanceand theamountofprimordial6 Li inferredfromobservations.© 2017 Elsevier B.V. All rights reserved.
∗ Corresponding author.
E-mail address: carlo.gustavino@roma1.infn.it (C. Gustavino).
1 Present address: Sächsisches Staatministerium für Umwelt und Landwirtschaft,
Dresden, Germany.
2 Present address: CPPM, Université d’Aix-Marseille, CNRS/IN2P3, Marseille,
France.
3 Deceased.
4 Present address: GSI, Darmstadt, Germany.
5 Present address: University of Warsaw, Warsaw, Poland. http://dx.doi.org/10.1016/j.astropartphys.2017.01.007
58 D. Trezzi et al. / Astroparticle Physics 89 (2017) 57–65
1.
Introduction
Big
Bang
Nucleosynthesis
(BBN)
may
be
used
to
probe
cosmo-logical
models
and
parameters.
To
this
end,
abundance
predictions
from
BBN
have
to
be
compared
with
the
abundances
inferred
by
astronomers
observing
the
emission
and
absorption
lines
in
spe-cific
astrophysical
environments.
The
pillars
of
BBN
calculations
are:
Standard
Cosmological
Big
Bang
model,
the
Standard
Model
of
Particle
Physics,
and
the
nuclear
cross
sections
of
the
processes
involved
in
the
BBN
reaction
network.
The
agreement
between
cal-culations
and
observations
for
primordial
deuterium
and
4He
(see
for
example
[1]
)
places
cosmology,
nuclear
and
particle
physics
in
a
uniquely
consistent
framework.
However,
the
situation
is
not
as
favorable
for
7Li
and
6Li
[2]
.
For
7Li,
the
predicted
abundance
is
about
a
factor
of
3
higher
than
the
observed
one.
As
of
today
a
clear
solution
to
this
puzzle,
“the
lithium
problem
”,
has
not
been
found.
Even
more
complex
is
the
case
of
6Li,
where
the
predicted
abundance
is
up
to
three
orders
of
magnitude
lower
than
that
in-ferred
from
direct
observations
(the
so
called
“second
lithium
prob-lem
”)
[3,4]
.
The
aim
of
the
present
work
is
to
put
the
nuclear
physics
of
6Li
production
in
standard
Big
Bang
scenarios
on
solid
experimental
ground.
The
nuclear
cross
section
of
the
leading
process
in
the
6Li
primordial
production,
i.e.
the
2H(
α
,
γ
)
6Li
fusion
reaction,
has
thus
been
directly
measured
in
the
BBN
energy
range
at
LUNA
(Labo-ratory
for
Underground
Nuclear
Astrophysics).
The
measurement
has
been
carried
out
with
the
world’s
only
underground
acceler-ator
for
nuclear
astrophysics,
situated
at
the
Laboratori
Nazionali
del
Gran
Sasso
(LNGS),
Italy.
The
data
at
center-of-mass
energy
E
=
133
and
93
keV
have
been
previously
published
in
abbreviated
form
[5]
.
Here,
two
new
data
points
at
E
=
120
and
80
keV
are
presented.
The
present
work
is
organized
as
follows:
In
Sections
2
and
3
a
description
of
astronomical
observations
and
nuclear
pro-cesses
involved
in
the
6Li
production
and
destruction
are
given.
In
Section
4
a
short
description
of
the
LUNA
apparatus
is
reported
(more
details
in
[6]
).
In
Section
5
data
analysis
is
described
and
finally
in
Sections
6
and
7
the
2H(
α
,
γ
)
6Li
reaction
rate
includ-ing
the
new
LUNA
data
points
and
the
resulting
conclusions
are
given.
2.
Review
of
astronomical
data
on
6Li
6
Li
is
mainly
produced
during
the
BBN
epoch
and,
in
more
recent
epochs,
by
cosmic
ray
spallation
[7]
.
For
this
reason,
its
primordial
abundance
is
inferred
from
observations
of
the
atmo-spheres
of
hot
metal-poor
stars
in
the
galactic
halo
(either
main
sequence
dwarfs
or
subgiants
near
the
turn-off point).
The
primor-dial
abundance
is
then
obtained
by
extrapolating
the
abundance
at
zero
metallicity.
The
strength
of
the
lithium
absorption
line
(
λ
=
670.7
nm)
provides
the
lithium
abundance.
As
the
absorption
line
of
6Li
is
slightly
shifted
towards
a
higher
wavelength
compared
to
7Li,
the
abundance
of
6Li
isotope
can
be
derived
through
the
shape
analysis
of
the
lithium
absorption
line.
The
shape
of
the
absorption
line
is
also
affected
by
the
convective
motions
of
the
stellar
atmo-sphere
thus
analysis
of
the
lithium
absorption
line
depends
on
the
theoretical
model
for
the
stellar
atmosphere.
Recently,
Steffen
et
al.
[10]
and
Lind
et
al.
[11]
have
pointed
out
that,
by
using
three-dimensional
model
atmospheres
and
dropping
the
assumption
of
Local
Thermodynamic
Equilibrium
(3D
NLTE),
many
of
the
6Li
detections
may
become
non-significant.
However,
the
situation
is
still
unclear.
As
an
example,
3D-NLTE
analysis
of
star
HD
84937
provides
contradictory
results
according
to
differ-ent
authors
[10,11]
.
In
other
cases,
such
as
star
G020-024,
we
have
a
good
agreement
between
one
dimensional
[3]
and
three
dimen-sional
models
[10]
,
respectively.
The
results
obtained
in
metal-poor
Table 1
Astronomical observations. Relative 6 Li/ 7 Li isotopic
abundance in some metal-poor stars obtained using dif- ferent stellar atmosphere models, in one (1D) or three (3D) dimensions, with or without Local Thermodynamic Equilibrium (LTE, NLTE), respectively.
Star name 6 Li/ 7 Li [%] method ref.
HD 84937 5 .0 ± 2.0 1D LTE [8] 5 .2 ± 1.9 1D LTE [9] 6 .6 ± 2.4 1D LTE [10] 5 .1 ± 1.5 1D NLTE [4] 5 .1 ± 2.3 3D NLTE [10] 1 .1 ± 1.0 3D NLTE [11] HD 160617 0 .1 ± 1.2 1D LTE [10] 0 .3 ± 1.2 1D LTE [10] 3 .6 ± 1.0 1D LTE/NLTE [3] −0.5 ± 1.2 3D NLTE [10] −1.3 ± 1.2 3D NLTE [10] −0.01 ± 0.01 3D NLTE [12] G064-012 5 .9 ± 2.1 1D NLTE [4] 2 .3 ± 2.9 1D LTE [10] 2 .0 ± 2.9 1D LTE [10] 1 .2 ± 2.8 3D NLTE [10] 0 .8 ± 2.7 3D NLTE [10] 0 .5 ± 2.3 3D NLTE [11] G020-024 11 .7 ± 2.5 1D LTE [10] 7 .0 ± 1.7 1D LTE [3] 9 .9 ± 2.4 3D LTE [10]
stars
(HD
84937,
HD
160617,
G064-012
and
G020-024)
where
dif-ferent
analyses
have
been
reported
in
the
literature,
are
shown
in
Table
1
.
In
conclusion,
although
several
authors
agree
on
the
fact
that
6
Li
has
been
detected
on
a
few
halo
stars
[2,10]
,
the
determination
of
primordial
6Li
abundance
is
still
a
difficult
task,
because
obser-vations
are
affected
by
systematic
uncertainties
related
to
the
con-vective
motion
of
the
stellar
atmosphere.
As
a
consequence,
obser-vations
with
spectrometers
with
higher
sensitivity
and
resolution
would
be
necessary.
At
the
same
time,
it
would
be
important
to
improve
stellar
modeling
in
order
to
firmly
establish
the
primor-dial
abundance
of
6Li
inferred
from
observations.
3.
The
nuclear
physics
of
primordial
6Li
In
the
standard
BBN
framework,
the
primordial
6Li
abundance
is
mainly
determined
by
two
nuclear
reactions
[13]
:
the
2H(
α
,
γ
)
6Li
reaction
that
produces
6Li
and
the
6Li(p,
α
)
3He
that
destroys
it.
The
6Li(p,
α
)
3He
reaction
rate
is
fairly
well
known
in
the
BBN
energy
range
[14]
.
On
the
other
hand,
the
lack
of
direct
measurements
at
low
energy
of
the
2H(
α
,
γ
)
6Li
process
makes
its
reaction
rate
largely
uncertain.
The
2H(
α
,
γ
)
6Li
cross
section
at
energies
less
than
1
MeV
is
dominated
by
radiative
E2
capture
from
d
waves
in
the
scattering
state
into
the
ground
state
of
6Li
through
a
3
+resonance
at
E
R=
711
keV.
At
energies
lower
than
300
keV
in
the
center-of-mass
sys-tem
system,
due
to
the
different
angular
momentum
barriers
for
p
and
d
waves,
the
E1
contribution
is
expected
to
be
comparable
to
the
E2
one
[15]
.
For
a
detailed
discussion
of
theoretical
predictions,
see
Refs.
[15,16]
.
The
2H(
α
,
γ
)
6Li
cross
section
has
never
been
directly
measured
in
the
BBN
region
of
interest
(30
࣠
E
࣠
300
keV),
before
the
LUNA
data
first
reported
in
[5]
.
It
is
worth
nothing,
only
one
attempt
to
directly
measure
the
2H(
α
,
γ
)
6Li
cross
section
at
low
energy
is
re-ported
in
literature,
providing
only
an
upper
limit
[17]
.
The
other
direct
measurements
have
been
performed
at
higher
energies,
i.e.
around
the
711
keV
resonance
[18]
,
and
in
the
MeV
region
[19]
.
In
order
to
unfold
the
2H(
α
,
γ
)
6Li
cross
section
at
low
energy,
two
dif-ferent
Coulomb
dissociation
experiments
have
been
performed
at
26
[20]
and
150
MeV/nucleon
[15]
projectile
energy,
respectively.
Fig. 1. Experimental setup. The 4 He + beam enters in the gas target through the
collimator (1), passes through the steel pipe (2) and reaches the calorimeter (3). The deuterium inlet (4) is below the calorimeter. The pressure is kept constant by means of a feedback system controlled by a pressure gauge (5). The
γ
-rays pro- duced by the 2 H(α
,γ
) 6 Li reaction are detected by a HPGe detector (6). The beaminduced background (see Section 4.2 for details), is monitored by a silicon detector (7). To reduce the environmental background, a lead castle (8) and an anti-radon box (9) surround the detector and gas target.
This
technique
is
mainly
sensitive
to
the
electric
quadrupole
(E2)
component
of
the
cross
section.
However,
the
competing
nuclear
breakup
process
dominates
at
both
beam
energies
[15,20]
.
There-fore,
these
data
may
be
interpreted
as
upper
limits
of
the
E2
con-tribution
to
the
cross
section.
In
conclusion,
the
6Li
abundance
pre-dicted
by
the
BBN
theory
is
affected
by
large
uncertainties
because
of
the
lack
of
data
for
the
2H
(
α
,
γ
)
6Li
reaction.
In
this
context,
the
LUNA
measurement
discussed
in
this
paper
represents
a
significant
step
forward,
providing
the
first
direct
measurement
of
this
reac-tion
well
inside
the
BBN
energy
region.
4.
Experimental
setup
4.1.
Apparatus
description
The
measurement
was
performed
by
detecting
the
prompt
γ
rays
emitted
from
the
2H(
α
,
γ
)
6Li
reaction,
using
the
experimen-tal
setup
shown
in
Fig.
1
.
A
4He
+beam
generated
by
the
400
keV
LUNA
II
accelerator
[21]
passed
several
collimators
along
a
pow-erful
gas
pumping
system
to
enter
the
target
chamber,
in
which
a
deuterium
gas
pressure
of
0.3
mbar
was
kept
by
a
feedback
system.
Together
with
the
good
energy
stability
of
the
accelerator,
the
win-dowless
gas
target
(length
17.7
cm)
allows
for
an
accurate
calcula-tion
of
the
beam
energy
along
the
entire
beam
path.
The
stopping
power
inside
the
gas
target
is
quite
low
(0.18
keV/cm
at
400
keV
beam
energy).
The
beam
intensity
(typically
0.3
mA)
was
measured
by
a
beam
calorimeter
with
a
constant
temperature
gradient.
The
target
chamber
and
HPGe
detector
were
shielded
by
at
least
20
cm
of
lead
in
all
directions
and
the
setup
was
enclosed
in
an
anti-radon
box
flushed
with
nitrogen
to
ensure
a
very
low
and
stable
natural
background
(see
Fig.
2
).
The
γ
-rays
were
de-tected
by
a
large
(137%
relative
efficiency)
high-purity
germanium
(HPGe)
detector
placed
at
a
90
° angle
with
respect
to
the
ion
beam
direction
in
a
close
geometry,
to
reach
a
full-energy
peak
detection
efficiency
of
1.7%
at
1.6
MeV.
An
energy
resolution
of
2.8
keV
at
1.6
MeV
γ
-ray
energy
was
achieved.
A
more
detailed
description
of
the
setup
can
be
found
in
[6,22]
.
4.2.
Neutron
induced
background
2
H(
α
,
γ
)
6Li
is
not
the
only
reaction
that
occurred
in
the
gas
tar-get.
Deuterons
can
be
elastically
scattered
by
the
incoming
alpha
Fig. 2. Black line: Natural background measured with a HPGe detector. Blue line: The 2 H(
α
,γ
) 6 Li spectrum at E α= 360 keV and ptarget = 0.3 mbar (natural back-
ground has been subtracted). Violet line: The 2 H(
α
,γ
) 6 Li simulated spectrum at Eα
= 360 keV and p target = 0.3 mbar (scaled down). The arrows indicate the (n,n’
γ
)lines. The triangular peaks are due to the interaction of neutrons with the ger- manium (see text). More details are reported in [6] . The red band indicates the
2 H(
α
,γ
) 6 Li region of interest at Eα= 360 keV. The measured beam induced back-
ground at LUNA is about one order of magnitude less than the natural background expected on Earth surface experiments with the same setup [24] . (For interpreta- tion of the references to colour in this figure legend, the reader is referred to the web version of this article.)
particles,
receiving
a
kinetic
energy
that
enables
them
to
induce
the
2H(
2H,p)
3H
and
2H(
2H,n)
3He
reactions
with
the
surrounding
deuterons
in
the
gas.
Protons
from
the
2H(
2H,p)
3H
reaction
were
monitored
with
a
silicon
detector
(see
Fig.
1
and
[6]
for
details).
Neutrons
from
the
2H(
2H,n)
3He
reaction
can
interact
with
the
sur-rounding
materials
(iron,
copper,
lead
and
germanium)
producing
γ
-rays
through
(n,n’
γ
)
reactions.
In
order
to
reduce
the
deuteron
scattering
probability,
a
pipe
with
a
square
section
of
2
cm
was
in-stalled
along
the
beam
line.
This
pipe
limits
the
free
path
of
the
elastically
scattered
deuterons
inside
the
gas,
thus
reducing
the
number
of
neutrons
produced
by
the
2H(
2H,n)
3He
reaction
to
a
value
of
about
10
neutrons
per
second.
The
rate
of
this
neutron
in-duced
background
BG
neutronin
the
Region
of
Interest
(RoI)
for
the
2H(
α
,
γ
)
6Li
reaction
is
about
one
order
of
magnitude
higher
than
the
expected
signal
but
nevertheless
much
lower
than
the
usual
background
present
on
Earth’s
surface
experiments
[23]
.
Fig.
2
shows
the
in-beam
E
α=
360
keV
γ
-ray
spectrum
as
well
as
a
comparative
background
spectrum.
It
contains
many
lines
and
several
triangular
shaped
peaks
(for
more
details
see
[6]
).
The
for-mers
are
due
to
the
natural
background
and
(n,n’
γ
)
beam
induced
reactions
on
the
materials
surrounding
the
reaction
chamber.
The
triangular
shaped
peaks
are
due
to
(n,n’
γ
)
beam
induced
reactions
of
energetic
neutrons
on
the
Germanium
detector
itself.
The
tri-angular
shape
arises
from
the
germanium
nucleus
recoil
energy
deposited
in
the
detector.
Similar
features
have
been
observed
at
all
beam
energies
investigated.
The
neutron
induced
background
spectrum
as
a
function
of
the
beam
energy
has
been
studied
with
a
dedicated
simulation,
in
which
our
experimental
setup
has
been
reproduced.
The
simulated
spectra
are
consistent
with
the
exper-imental
ones
[22]
.
The
main
difference
between
the
spectra
ob-tained
at
different
beam
energies
(experimental
and
simulated),
is
an
overall
energy
dependence
of
the
HPGe
response
at
E
γ࣡
1500
keV,
in
such
a
way
the
bin-by-bin
ratio
produced
at
dif-ferent
beam
energies
is
not
constant
but
weakly
depends
on
the
beam
energy.
For
what
concerns
the
complex
spectral
structure,
the
most
prominent
difference
between
spectra
at
different
beam
60 D. Trezzi et al. / Astroparticle Physics 89 (2017) 57–65 0 500 1000 1500 2000 2500 3000 3500 4000 4500 1450 1500 1550 1600 a.u. Energy [keV] isotropic emission pure E1 emission pure E2 emission
Fig. 3. Simulated spectra of the 2 H(
α
,γ
) 6 Li reaction (E α= 400 keV), assuming dif-ferent angular distribution for the emitted photons. Note that the RoI does not de- pends on the adopted angular distributions.
energies
is
related
to
the
excitation
of
56Fe
nuclei
to
the
2658
keV
state,
which
decays
in
a
cascade
via
emission
of
1811
keV
and
847
keV
γ
-rays.
The
strength
of
the
gamma
lines
associated
to
the
2658
keV
state
increases
with
the
beam
energy.
In
fact,
at
high
energy
(360
or
400
keV)
there
are
more
neutrons
available
to
pop-ulate
the
2658
keV
level
with
respect
to
low
beam
energies
(280
or
240
keV),
thus
explaining
the
increase
of
the
1811
keV
counting
rate
with
the
beam
energy.
This
effect
is
confirmed
in
a
dedicated
simulation
in
which
the
Rutherford
α
(
2H,
2H)
α
scattering
and
the
2H(
2H,n)
3He
reaction
are
implemented.
As
expected,
it
has
been
found
that
the
doppler
effect
determines
a
broadening
of
neutron
energy
distribution
with
the
beam
energy,
in
agreement
with
the
observed
effect
on
the
56Fe
gamma
lines
[6,22]
.
5.
Data
analysis
5.1.
General
approach
In
this
section
the
data
analysis
of
the
2H(
α
,
γ
)
6Li
reaction
at
beam
energies
240,
280,
360
and
400
keV
is
reported.
The
energy
of
photons
produced
in
the
2H(
α
,
γ
)
6Li
reaction
depends
on
the
beam
energy
and
on
the
doppler
effect.
It
can
be
expressed
by
the
following
relativistic
formula
(in
which
c
=
¯h
=
1
):
E
γ=
m
2 He+
m
2d− m
2 Li+
2
m
d(
E
α+
m
He)
2
[
E
α+
m
He+
m
d− p
Hecos
(
θ
lab)
]
(1)
In
this
equation
E
γis
the
photon
energy,
m
He,
m
dand
m
Liare
the
masses
of
nuclides
involved
in
the
reaction,
p
He=
E
α(
E
α+
2
m
He)
is
the
α
particle
momentum
and
θ
labis
the
angle
of
emitted
photon
with
respect
to
the
beam
axis.
This
formula
indicates
that
fully
detected
photons
populate
a
well
defined
region
of
interest
(RoI)
for
each
beam
energy.
The
energy
range
of
the
RoIs
only
depends
on
the
beam
energy
and
on
the
angular
acceptance
of
the
HPGe
detector
(10
°
࣠
θ
lab࣠
170
°).
It
has
been
determined
with
the
GEANT
Monte
Carlo
sim-ulation,
in
which
the
experimental
conditions
are
reproduced
and
the
Eq.
(1)
is
implemented.
The
angular
distribution
of
photons
produced
by
the
2H(
α
,
γ
)
6Li
only
affects
the
distribution
of
fully
detected
photons
inside
the
RoI.
Fig.
3
shows
the
simulated
en-ergy
spectra
for
different
angular
distribution
of
emitted
pho-tons.
The
fully
detected
photons
produced
by
the
2H(
α
,
γ
)
6Li
re-action
have
the
following
RoIs:
[1542,1563]
keV,
[1554,1577]
keV,
[1580,1606]
keV
and
[1592,1620]
keV,
respectively
for
beam
ener-gies
of
240,
280,
360
and
400
keV.
We
developed
two
independent
analysis
methods.
Our
data
have
been
grouped
in
pairs
(i,j)
of
different
beam
energy
runs
char-acterized
by
non-overlapping
RoIs,
comparable
statistics
and
simi-lar
integrated
beam
current.
If
we
now
indicate
with
BG
neutroni
and
BG
neutronj
the
rate
of
neutron
induced
background
relative
to
i
and
j
energies,
we
have:
BG
neutroni
=
β
i jBG
neutronj(2)
As
it
has
been
discussed
in
Section
4.2
,
the
shape
of
beam
in-duced
background
weakly
depends
on
the
beam
energy.
In
this
for-mula
this
dependence
is
included
in
the
β
ijparameter
and
will
be
considered
in
the
following.
Both
analysis
methods
provide
cross
section
values
σ
(
E
),
pa-rameterized
by
the
astrophysical
S-factor
given
by:
S
(
E
)
=
σ
(
E
)
E
exp
72
.
44
/
√
E
(3)
where
E
(
keV
)
is
the
center-of-mass
system
energy.
The
two
analy-sis
methods
named
“Flat
regions” and
“Minimization” will
now
be
described
in
detail.
5.1.1.
Method
A,
based
on
“flat
regions”
After
subtraction
of
the
natural
background,
there
are
several
energy
regions
without
any
visible
peak.
This
purely
neutron-induced
background
is
mainly
due
to
the
Compton
continua
of
several
peaks
in
the
γ
-ray
spectrum.
Therefore,
it
is
possible
to
parameterize
the
ratio
of
background
rates
inside
such
“flat
re-gions” at
two
beam
energies
along
the
entire
γ
-ray
spectrum.
In
this
way,
the
ratio
of
background
levels
can
be
calculated
anywhere
in
the
γ
-ray
spectrum.
This
allows
to
properly
subtract
beam
in-duced
backgrounds
obtained
at
two
different
beam
energies
using
as
normalization
factor
the
background
ratio.
Given
that
the
two
RoIs
do
not
overlap,
The
two
signals
Y
Enetα=400 keV(RoI
400
keV)
and
Y
Enetα=280 keV(RoI
280
keV),
can
be
written
as
Y
400 net(
400
)
=
Y
400(
400
)
−
β
400,280Y
400(
280
)
(4)
Y
280 net(
280
)
=
Y
280(
280
)
−
1
β
400,280Y
280(
400
)
(5)
Similar
equations
can
be
written
for
the
other
two
beam
en-ergies
(E
α=
240
and
360
keV).
The
systematic
uncertainty
com-ing
from
the
determination
of
the
normalization
factor
(here
called
β
ROI400,280
and
β
360ROI,240) from neighboring “flat regions” is difficult to
estimate.
Therefore,
instead
of
deriving
the
systematic
error
bar
of
β
ROI400,280
,
an
approach
is
chosen
that
cancels
its
effects
on
the
sig-nal
yield,
and
thus
on
the
final
astrophysical
S-factor.
It
is
clear
from
Eqs.
(4)
and
(5)
that
β
ROI400,280
uncertainties
affect
the
two
yields
in
opposite
ways:
If
β
ROI400,280
is
overestimated,
Y
net400(
400
)
de-creases
but
Y
280net
(
280
)
increases.
However,
we
found
that
the
yield
sum
Y
400net
(
400
)
+
Y
net280(
280
)
is
independent
from
β
400ROI,280,
so
that
the
issue
of
the
error
estimation
of
this
parameter
becomes
moot.
To
extract
the
astrophysical
S-factor
at
a
given
energy
from
the
yield
sum
Y
400net
(
400
)
+
Y
net280(
280
)
,
it
is
necessary
to
make
an
as-sumption
on
the
energy
dependence
of
the
astrophysical
S-factor
inside
the
LUNA
energy
range
[25]
.
For
the
data
analysis,
various
assumptions
ranging
from
S
(400
keV)/
S
(280
keV)
=
1.1–1.5
have
been
tested,
where
1.1
corresponds
to
pure
E1
and
1.5
to
pure
E2.
The
S-factor
result
does
not
change
outside
its
statistical
error
bar
in
any
of
the
cases
tested.
The
Method
A
analysis
results
are
shown
in
Table
2
,
where
the
theoretical
S-factor
trend
indicated
in
[15]
is
assumed,
with
a
ratio
S
(400
keV)/
S
(280
keV)
=
1.3.
In
summary,
in
method
A
the
effects
of
the
background
subtrac-tion
cancel
out,
so
the
result
does
not
depend
on
the
uncertainty
of
the
background
subtraction.
However,
this
method
requires
a
the-oretical
assumption
for
the
shape
of
the
S-factor
curve,
and
several
Table 2
Astrophysical S-factor and nuclear cross section. In the table are indicated: the effective center-of-mass energy E [keV], the beam energy E α[keV], the deuterium gas target pressure P [mbar], the measurement time t [h], the integrated beam current Q [C] =
I beam t, the net reaction yield Y [counts/C] and its statistical error (the counting error for the “method A” includes the uncertainty
due to the theoretical assumption, see text), the total cross section
σ
[pb] with counting and systematic errors, the astrophysical S-factor S [keVμ
b] with counting and systematic errors. The results of analysis “Method B” are adopted in this work. The data points at 80 and 93 keV are less than two standard deviations above zero and should be considered as upper limits.E [keV] E α[keV] P [mbar] Q [C] t [h] Yield [counts/C] cross section [pb] S-factor [keV
μ
b] Method 80 240 0 .306 211 .5 217 .7 0 .23 ± 0.64 9 ± 25 ± 1 2 .4 ± 6.7 ± 0.3 A 93 280 0 .308 538 .9 490 .9 0 .46 ± 0.46 18 ± 18 ± 2 3 .0 ± 3.0 ± 0.4 A 120 360 0 .306 252 .7 205 .2 0 .94 ∓ 0.57 37 ∓ 23 ± 5 3 .3 ∓ 2.0 ± 0.4 A 133 400 0 .306 514 .3 437 .7 1 .46 ∓ 0.46 58 ∓ 18 ± 8 4 .1 ∓ 1.3 ± 0.5 A 80 240 0 .306 211 .5 217 .7 0 .11 ± 0.51 4 ± 20 ± 0 .4 1 .1 ± 5.2 ± 0.1 B 93 280 0 .308 538 .9 490 .9 0 .42 ± 0.25 16 ± 10 ± 2 2 .7 ± 1.6 ± 0.3 B 120 360 0 .306 252 .7 205 .2 1 .05 ± 0.47 40 ± 18 ± 6 3 .6 ± 1.6 ± 0.5 B 133 400 0 .306 514 .3 437 .7 1 .50 ± 0.32 59 ± 13 ± 7 4 .1 ± 0.9 ± 0.5 Bassumptions
have
been
tested.
Even
still,
a
radically
different
S-factor
curve
cannot,
in
principle,
be
excluded.
An
alternative
anal-ysis
method
to
derive
the
cross
section
without
any
theoretical
as-sumption
on
the
shape
of
the
S-factor
curve
is
described
in
the
next
section.
5.1.2.
Method
B,
based
on
minimization
procedures
This
method
is
based
on
Eq.
(2
).
The
γ
-ray
spectrum
rate
R
iis
composed
by
the
neutron-induced
background
BG
neutroni
,
the
natu-ral
background
BG
roomand
the
2H(
α
,
γ
)
6Li
counting
rate.
This
latter
can
be
expressed
as
k
iN
i,
in
which
k
iis
a
scaling
pa-rameter
and
N
iis
the
energy
distribution
due
to
the
photons
pro-duced
by
the
2H(
α
,
γ
)
6Li
reaction.
The
N
i
distribution
has
been
ob-tained
with
the
dedicated
GEANT
simulation
in
which
all
the
de-tails
of
the
experimental
conditions
are
reproduced
and
2H(
α
,
γ
)
6Li
reactions
are
generated
along
the
beam
axis
(see
Fig.
3
).
The
BG
neutronrate
can
be
written
as
follows:
BG
neutroni
=
R
i− BG
room− k
iN
i(6)
Substituting
Eq.
(6)
in
Eq.
(2)
we
obtain:
R
i− BG
room− k
iN
i−
β
i jR
j− BG
room− k
jN
j=
0
(7)
With
the
“Method
B” the
free
parameters
β
ij,
k
iand
k
jin
the
Eq.
(7)
are
obtained
with
a
MINUIT
χ
2(least
squares)
minimiza-tion
procedure
in
the
energy
window
[1500,1625]
keV,
which
in-cludes
all
the
considered
RoIs.
As
stated
above,
the
shape
of
the
distribution
of
fully
detected
photons
inside
a
RoI
strongly
depends
on
the
adopted
angular
distribution
of
emitted
photons
(see
Fig.
3
).
In
this
concern,
all
the
counting
levels
inside
of
the
RoI’s
(natural
background,
beam
induced
background
and
signals)
have
been
averaged
before
the
minimization
procedure,
to
make
analysis
sensitive
only
to
the
to-tal
cross
section
and
not
dependent
on
angular
distribution
as-sumptions.
The
HPGe
efficiency
for
different
angular
distributions
has
been
tested
with
the
simulation.
As
a
result,
the
efficiency
un-certainty
due
to
the
unknown
angular
distribution
has
been
con-servatively
quoted
at
the
9%
level.
As
discussed
in
Section
4.2
,
the
β
ijparameter
weakly
depends
on
the
energy.
This
dependence
has
been
modeled
with
a
second
order
polynomial
in
our
minimization
region.
This
correction
de-termines
a
negligible
variation
of
the
cross
sections,
about
a
fac-tor
four
lower
than
the
statistical
errors,
for
all
the
four
consid-ered
data
points.
As
discussed
in
Section
4.2
,
the
strength
of
the
1811
keV
line
slightly
depends
on
the
beam
energy.
Although
the
line
is
outside
the
minimization
interval,
its
Compton
edge
is
at
about
1580
keV.
To
evaluate
its
effect
in
the
analysis
results,
this
line
and
its
Compton
edge
have
been
subtracted
from
the
exper-imental
spectra.
To
do
so,
the
strength
of
the
1811
keV
peak
of
each
dataset
has
been
used
to
normalize
and
subtract
the
simu-lated
spectrum.
This
correction
was
found
to
be
negligible
as
well,
about
a
factor
20
lower
than
the
statistical
error,
for
all
the
four
energies
considered.
The
results
of
Method
B
described
above
are
shown
in
Table
2
.
The
analysis
has
been
repeated
using
wider
en-ergy
windows
for
the
minimization
procedure,
up
to
500
<
E
<
2500
keV,
obtaining
results
fully
consistent
with
those
presented
in
Table
2
.
As
a
consistency
check,
the
minimization
procedure
has
been
repeated
in
energy
windows
not
containing
the
RoI’s.
As
ex-pected,
the
counting
difference
between
paired
spectra
is
always
consistent
with
zero,
within
statistical
fluctuations
[26]
.
6.
Results
and
astrophysical
aspects
6.1.
Astrophysical
S-factor
and
nuclear
cross
section
The
relevant
measurement
parameters
and
the
results
of
the
two
analysis
are
reported
in
Table
2
.
The
yield
Y
iinside
a
RoI
is
defined
by
the
following
formula:
Y
i=
k
in
RoIit
i/Q
i(8)
where
n
RoIi
are
the
net
counts/sec
in
the
RoI,
t
iis
the
acquisition
time.
In
the
“Method
A” the
error
of
Y
ireported
in
Table
2
is
due
to
the
statistical
fluctuations
inside
the
two
paired
RoI’s
and
to
the
uncertainties
of
the
theoretical
assumption
adopted
to
estab-lish
the
energy
trend
of
the
cross
section.
The
yield
uncertainty
obtained
with
“Method
B” has
been
obtained
in
the
χ
2(least
squares)
minimization
procedure
of
MINUIT,
in
which
the
corre-lation
between
β
,
k
i,
and
k
jis
taken
into
account
by
means
of
co-variance
matrix.
It
is
essentially
due
to
the
counting
fluctuations
and
therefore
it
determines
the
statistical
significance
of
the
ob-tained
results.
Note
that
the
significance
of
Y
400net
(
400
)
is
more
than
4
σ
,
while
Y
240net
